A170780 a(n) = n^8*(n^8 + 1)/2.

0, 1, 32896, 21526641, 2147516416, 76294140625, 1410555793536, 16616468167201, 140737496743936, 926510115949281, 5000000050000000, 22974865038965521, 92442129662509056, 332708304999455281, 1088976669642580096

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 16 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=32896, there are 2^16=65536 oriented arrangements of two colors. Of these, 2^8=256 are achiral. That leaves (65536-256)/2=32640 chiral pairs. Adding achiral and chiral, we get 32896. - Robert A. Russell, Nov 13 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Crossrefs

Row 16 of A277504.

Cf. A010804 (oriented), A001016 (achiral).

Programs

GAP

List([0..30], n -> n^8*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

Magma

[n^8*(n^8+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 26 2011

Mathematica

Table[n^8*(n^8+1)/2, {n, 0, 30}] (* G. C. Greubel, Dec 05 2017 *)

PARI

for(n=0, 30, print1(n^8*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 05 2017

Python

for n in range(0,20): print(int(n**8*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

Sage

[n^8*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

Formula

G.f.: (x + 32879*x^2 + 20967545*x^3 + 1786036695*x^4 + 42691617829* x^5 + 391057805899*x^6 + 1603741496717*x^7 + 3191399514435*x^8 + 3191399514435*x^9 + 1603741496717*x^10 + 391057805899*x^11 + 42691617829*x^12 + 1786036695*x^13 + 20967545*x^14 + 32879*x^15 + x^16) /(1-x)^17. - G. C. Greubel, Dec 05 2017

From Robert A. Russell, Nov 13 2018: (Start)

a(n) = (A010804(n) + A001016(n)) / 2 = (n^16 + n^8) / 2.

G.f.: (Sum_{j=1..16} S2(16,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x*Sum_{k=0..15} A145882(16,k) * x^k / (1-x)^17.

E.g.f.: (Sum_{k=1..16} S2(16,k)*x^k + Sum_{k=1..8} S2(8,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>16, a(n) = Sum_{j=1..17} -binomial(j-18,j) * a(n-j). (End)